We have for a convex body in H² established L(g) = ò^2p₀ h_E(j)dj where h_E is the Euclidean distance in the Beltrami-Klein model to the right-hand support line, our Projective Cauchy formula. We obtain the polar form, L(g) = ò^2p₀ sinh²r[dw/ ds] dq ş ò^2p₀ K_g [(sinh²rcoshr)/(cosh²r)] dq in the smooth case (which is easily adapted to the convex case). This formula is intrinsic and adapts to S² (and of course, E²) immediately. We contrast this and equivalent characterizations, some classical, with our non-equivalent (saving E²) Minkowski form, L(g) = ò^2p₀ K_g([(L(r))/(2p)])² dq+ K ò_g[(A(r))/(2p)] ds, where L(r), A(r) are respectively circumference and included area of circle of radius r9